Simplify and expand the following expression: $ \dfrac{4p + 7}{4p - 7}+\dfrac{2p}{p + 10} $
Solution: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(4p - 7)(p + 10)$ Multiply the first term by $\dfrac{p + 10}{p + 10}$ $ \begin{align*} \dfrac{4p + 7}{4p - 7} \times \dfrac{p + 10}{p + 10} & = \dfrac{(4p + 7)(p + 10)}{(4p - 7)(p + 10)} \\ & = \dfrac{4p^2 + 47p + 70}{(4p - 7)(p + 10)}\end{align*} $ Multiply the second term by $\dfrac{4p - 7}{4p - 7}$ $ \begin{align*} \dfrac{2p}{p + 10} \times \dfrac{4p - 7}{4p - 7} & = \dfrac{(2p)(4p - 7)}{(p + 10)(4p - 7)} \\ & = \dfrac{8p^2 - 14p}{(p + 10)(4p - 7)}\end{align*} $ Now we have: $ = \dfrac{4p^2 + 47p + 70}{(4p - 7)(p + 10)} + \dfrac{8p^2 - 14p}{(p + 10)(4p - 7)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{4p^2 + 47p + 70 + 8p^2 - 14p}{(4p - 7)(p + 10)} $ $ = \dfrac{12p^2 + 33p + 70}{(4p - 7)(p + 10)}$ Expand the denominator: $ = \dfrac{12p^2 + 33p + 70}{4p^2 + 33p - 70}$